Feed water, which is water with impurities, can be purified in a household or other local environment, by pressurizing the feed water to force it downstream through a filter arrangement. The resulting filtrate flows into a filtrate storage region. The filter arrangement may comprise a bundle of tubes of filtering material with water passing through the tube walls between the tube outside and inside. The tube material has pores that become clogged with microscopic particles during filtering. The filter tubes can be unclogged by backflushing, wherein a portion of the filtrate is forced in a reverse direction, or upstream direction, through the filtering material, with the backflushed water dumped into a drain. One type of filtering system includes a stretchable bladder that lies in the filtrate storage region, with filtrate on one bladder face and with pressured air on the other bladder face. As filtrate enters the filtrate storage region, the membrane deflects and compresses the air. The pressured air will later press the filtrate in an upstream direction during the next backflushing. One problem encountered with this arrangement is that the backflushing pressure may be lower than desirable, and decreases as backflushing proceeds. Another problem encountered is that the pressure differential across the filter arrangement may be lower than desirable, and decreases as filtering proceeds and bladder air pressure increases. A filtering system that produced constantly high pressure during filtering and during backflushing, would be desirable. It also would be desirable if a backflush pressure could be applied that was even higher than the pressure applied during filtrate production.